Alternating quotients of the (3.q.r) triangle groups

Abstract

A long standing conjecture (attributed to Graham Higman) asserts that each of the triangle groups Delta(p, q, r) For 1/p + 1/q + 1/r < 1 contains among its homomorphic ima all but finitely many of the alternating or symmetric groups. This phenomenon has been termed property H by Mushtaq and Servatius [9]. The work of several authors over the last decade and a half has shown that for any value of q, there are only finitely many r such that Delta(2,q, r) fails to have property H. In this paper, the techniques used by these authors ale generalised to handle the possibility that p is odd, and as a result. it is shown that for any q greater than or equal to 3, there are only finitely many r such that Delta(3, q, r) fails to have properly H.